Dynamic Financial Hedging Strategies for a Storable
Commodity with Demand Uncertainty
Panos Kouvelis
Olin School of Business, Washington University in St. Louis, US, [email protected]
Zhan Pang
Department of Management Science, Lancaster University, Lancaster, United Kingdom [email protected]
Qing Ding
Lee Kong Chian School of Business, Singapore Management School, Singapore [email protected]
We consider a firm purchasing and processing a storable commodity in a volatile commodity price market.
The firm has access to both a commodity spot market and an associated financial derivatives market. The
purchased commodity serves as a raw material which is then processed into an end product with uncertain
demand. The objective of the firm is to coordinate the replenishment and financial hedging decisions to
maximize the mean-variance utility of its terminal wealth over a finite horizon. We employ a dynamic
programming approach to characterize the structure of optimal time-consistent policies for inventory and
financial hedging decisions of the firm. Assuming unmet demand is lost, we show that under forward hedges
the optimal inventory policy can be characterized by a myopic state-dependent base-stock level. The optimal
hedging policy can be obtained by minimizing the variance of the hedging portfolio, the value of excess
inventory and the profit-to-go as a function of future price. In the presence of a continuum of option strikes,
we demonstrate how to construct custom exotic derivatives using forwards and options of all strikes to
replicate the profit-to-go function. The financial hedging decisions are derived using the expected profit
function evaluated under the optimal inventory policy. These results shed new light into the corporate risk
and operations management strategies: inventory replenishment decisions can be separated from the financial
hedging decisions as long as forwards are in place, and the dynamic inventory decision problem reduces to
a sequence of myopic optimization problems. In contrast to previous results, our work implies that financial
hedges do affect optimal operational policies, and inventory and financial hedges can be substitutes. Finally,
we extend our analysis to the cases with backorders, price-sensitive demand, and variable transaction costs.
Key words : Finance-Operations Interface, commodity price risk, inventory management, financial hedging.
History : May 2013
1.
Introduction
Commodity price volatility has a significant impact on raw material purchasing cost and inventory
management in many industries. For example, in chemical industry, manufacturers’ (e.g., BASF)
cost structures largely depend on the price of crude oil which experienced tremendous fluctuations
in the past several years, ranging from around $30 in 2003 up to a July 2008 high of $147.30 but then
down to a December 2008 low of $32 (Berling and Martı́nez-de-Albéniz 2011). Managing commodity
1
2
material price risk has become one of the biggest challenges for supply chain management. More and
more firms have realized the strategic importance of integrating the financial risk management and
operational management in their supply chains. For example, in December 2001, Ford announced
a loss of $1 billion on its precious-metals inventory and forward-contract agreements. On the
contrary, HP was able to cut its purchasing price risk for flash memory using its procurement risk
management strategy based on a contract portfolio approach (Nagali et al. 2008). The survey of
Bodnar et al. (1998) shows that 50% of U.S. non-financial firms were using financial derivatives as
risk management instruments.
There is a growing research interest in studying the financial risk management in supply chain
and operational contexts. We refer to Kleindorfer (2008a,b) for excellent overviews of the recent
developments and the research challenges to use financial instruments (e.g., forwards, futures and
options) in production and supply chain management. Our paper adds to this literature by addressing a joint dynamic financial hedging and inventory control problem for a risk-averse firm with
mean-variance utility of the terminal wealth. We attempt to answer the following questions: (1)
How to find a time-consistent integrated inventory and financial hedging policy under the meanvariance framework? (2) What is the structure of the optimal joint replenishment and financial
hedging policy? (3) What is the distinguishing role of inventory and financial hedges in managing
commodity price and demand risks? and (4) Are inventory and financial hedges complements or
substitutes?
More specifically, we consider a firm purchasing a storable commodity from a volatile (physical)
spot market with the access to associated financial derivatives. This commodity serves as a raw
material which is then processed into an end product with fixed selling prices. The demand of the
end product is random and depends on spot price. Unmet demand is treated as lost sales. The spot
price evolves as a Markov process. The physical spot market and the related financial derivatives
market (with the financial instruments written on the spot price) are assumed to be liquid in that
no transaction cost is incurred. Moreover, we assume that there is no arbitrage opportunity in
both the physical spot market and financial market. The objective of the firm is to maximize the
mean-variance utility of terminal wealth.
We formulate this problem as a stochastic dynamic program. Note that the mean-variance criterion is not time consistent which will induce a decision maker to deviate from the previous decision
at a later time. This is because the variability of terminal wealth that the decision maker perceives
at any point of time is higher than that of a later date. Inspired by Basak and Chabakauri (2010),
we apply the law of total variance (Weiss 2005) to develop the recursive equation by decomposing
the conditional mean-variance utility at any date into two parts. The first part is the expected
future mean-variance utility and the second the conditional variance of expected future cash flow.
3
The second part serves as an additional risk adjustment due to the greater perceived variability
compared to the variability that will be perceived in next period, which provides an incentive to
deviate from the decisions made in previous periods. Note that although the mean-variance criterion is not time-consistent, the derived policy in our model is time-consistent, namely, the optimal
decisions in the future are also optimal now.
We analyze two types of financial hedging strategies. We first assume that only the vanilla
derivative contracts such as forwards and simple options (e.g., European puts and calls) are used.
We show that, in the presence of a forward contract maturing at next period, the joint replenishment
and hedging optimization problem can be decomposed into two sub-problems: One is a single-period
inventory decision problem and the other involves the financial hedging decisions. Therefore, myopic
state-dependent base-stock policies are optimal. On the other hand, the dynamic financial hedging
decisions can be obtained by minimizing the variance associated with the expected future cash
flow conditioned on the next-period spot price. We then show how to construct exotic derivative
contracts with European calls and puts of all strikes to fully replicate the expected future cash
flow conditioned on the next period’s price. Note that exotic derivatives refer to all those nonstandard and tailor-made derivative contracts which normally cannot be replicated by a finite
number of standard exchange-traded vanilla derivatives, and mostly traded in over-the-counter
(OTC) markets (Briys et al. 1998). Thus the continuum of option strikes also belongs to the exotic
derivatives.
In order to examine whether the insight applies to more general settings, we study several
important extensions: (1) demand backordering; (2) price-sensitive demand; and (3) transaction
costs. When demand is backlogged, we show that the replenishment decisions are still separable
from the financial hedging decisions, while the hedging decision must take into account the optimal
pricing and inventory policies when predicting future cash flows. When demand is price-sensitive,
we show that the joint pricing and replenishment decisions can still be made myopically and
separated from financial hedging decisions. When there exist variable transaction costs, the myopic
policies are no longer optimal, but we can still use the exotic contracts to replicate the profit-to-go,
which implies that the replenishment decisions can still be separated from the financial hedging
decisions.
Our contributions to the literature are two-fold. First, we develop a dynamic programming
framework to derive the time-consistent optimal inventory and financial hedging policies under
an inter-period mean-variance utility. Second, our analysis sheds new light into effective commodity risk management practices: Inventory is used to hedge demand (quantity) risk while financial
derivatives are used to hedge the price risk. The inventory replenishment decisions can be separated from financial hedging decisions if the portfolio of hedging derivatives is structured properly.
4
Financial hedging does affect inventory policies and often simplifies them. However, optimal financial hedging strategies are heavily dependent on inventory policies as they determine future cash
flows. In other words, inventory decisions can be separated from financial hedging decisions, but
not the other way around.
This paper is organized as follows. Section 2 briefly reviews the related literature. Section 3
presents the model and the dynamic programming formulation. Section 4 characterizes the optimal
joint replenishment and hedging policies under different hedging strategies. Sections 5, 6 and 7
discuss the relationship between financial hedges and inventory hedges, elaborate on how financial
hedges create value, and illustrate numerically some of the obtained insights. Section 9 concludes
the paper. All the proofs are in the Appendix.
2.
Related Literature
This section reviews the related literature. We first briefly review the most relevant financial literature and then focus on the operations management literature.
Starting from the seminar paper of Markowitz (1952), the mean-variance analysis has long been
seen as the cornerstone of modern portfolio theory and has been widely adopted in both academia
and industry. There is an extensive literature about the mean-variance analysis of portfolio optimization. We refer to Basak and Chabakauri (2010) for a comprehensive review of its recent
developments. In a complete market, it has been shown that a (static) pre-commitment portfolio
strategy chosen at an initial date also solves the one with a quadratic objective for some specific parameters (e.g., Li and Ng 2000). However, Basak and Chabakauri (2010) point out that an
investor may subsequently find it optimal to deviate from the initial policy if market conditions
change in the future. They solve a continuous-time dynamic asset allocation problem of a meanvariance optimizer in a complete market setting and provide a simple dynamic portfolio policy.
Their work appears to be the first to obtain in an incomplete market environment a closed-form
solution of the dynamically optimal policy, from which the investor has no incentive to deviate,
namely, time-consistent policies. In a similar setting, Basak and Chabakauri (2011) study how to
hedge a non-tradable asset with a tradable asset. They obtain time-consistent hedges by dynamic
programming and show that their dynamically optimal hedges typically outperform their static
and myopic counterparts. Inspired by the developments of Basak and Chabakauri (2010, 2011),
we develop a discrete time dynamic programming framework to address the joint replenishment
and financial hedging problem. Our model differs from their models mainly in two aspects: (1) We
explicitly incorporate and address demand uncertainty. (2) We consider the integration of dynamically optimal operational (inventory) hedges and financial hedges, while there are no operational
decisions in their model. (3) We consider a discrete time setting as opposed to their continuous
5
time model, which allows us to decompose the inventory and financial hedging decisions and better
explore the structural insights.
Our work also relates to the optimal positioning problem in derivative securities. Carr and Madan
(2001) show that a twice-differentiable payoff function can be replicated by a risk-less asset (the
bond), a single risky asset (the stock) and European options of all strikes (derivatives). Oum et al.
(2006) employ Carr and Madan’s (2001) approach to develop the optimal static hedging strategy in
a competitive electricity wholesale market when both demand and price are uncertain. Our model
further extends Carr and Madan’s (2001) analysis to construct the dynamic replication strategies
in an inventory system.
Our paper is in line with the literature on the joint operational and financial risk management;
see Kleindorfer (2008a,b), Zhu and Kapuscinski (2011) and Kouvelis et al. (2013) for reviews of
recent developments. Compared to the financial literature, there is relatively less attention that
has been paid to the financial risk management in operational environments. Although some of
the tools in finance, such as risk averse decision criterion (e.g., mean-variance analysis of Sobel
1994, expected utility framework of Bouakiz and Sobel 1992, Eeckhoudt et al. 1995 and Agrawal
and Seshadri 2000), portfolio optimization (e.g., Martı́nez-de-Albéniz and Simchi-Levi 2005) and
financial option theory (e.g., Ding et al. 2007), have been introduced into operations research,
they do not take into account the unique aspects of commodity risk management in uncertain
demand environments. Next, we review the related literature in the single-period (newsvendor)
and multi-period inventory setting, respectively.
Recent developments on the interfaces of operational and financial risk management mainly focus
on the single-period (newsvendor) setting. Gaur and Seshadri (2005) address a joint inventory
ordering and financial hedging problem for a retailer with demand being correlated to the price
of a financial asset. They first show how to construct single-period hedging strategies in both
the mean-variance and the more general utility-maximization frameworks. They then investigate
the impact of financial hedges on the expected utility and the inventory levels. They show that
financial hedging increases the optimal inventory level for the risk averse newsvendor, and brings
it closer to the risk-neutral profit-maximization decision. With weather sensitive demand, Gao
et al. (2011) study a joint inventory ordering and weather derivative hedging decision problem
under a CVaR framework and obtain similar insights. Ding et al. (2007) consider the integration
of operational and financial hedging strategies for a global firm facing exchange rate risk. Chod
et al. (2010) examine the relationship between the operational flexibility and financial hedging in
capacity investment decisions. They show that product flexibility and financial hedging tend to be
complements (substitutes) when product demands are positively (negatively) correlated, whereas
postponement flexibility is a substitute to financial hedging. This literature has also been extended
6
to supply chain settings (e.g., Caldentey and Haugh 2009). Caldentey and Haugh (2009) address
the impact of financial hedging on the supply chain performance and contract design, and show that
in a typical two-stage supply chain the supplier always prefers the flexible contract with financial
hedging to the flexible contract without financial hedging whereas the buyer might or might not
prefer the flexible contract with hedging.
There are only a few papers addressing the joint financial hedging and operational decisions in
multi-period settings. With a quadratic utility function, Caldentey and Haugh (2006) provide a
modeling framework that allows continuous trading in the financial market to hedge against a cash
flow driven by an operational policy. With an additive exponential utility function, Smith and Nau
(1995) address a project valuation problem and show that the operational and financial hedging
decisions can be separated from each other. Applying the same framework of Smith and Nau (1995),
Chen et al. (2007) propose a multi-period consumption model that integrates pricing, inventory
control, consumption and financial hedging decisions. Assuming that financial hedging decisions
are made after observing the realized demand, they show that the inventory and pricing decisions
can be separated from the consumption and financial hedging decisions. Their results indicate
that the financial hedging does not change the inventory and pricing decisions when compared to
the system without financial hedges. In a similar setting, Geman and Ohana (2008) address the
problem of managing a storable commodity portfolio that includes physical assets and hedging
positions in spot and forward markets. Their discussion focuses on the time-consistency of the
utility functions. Note that the utility functions used by Chen et al. (2007) and Geman and Ohana
(2008) are both intra-period (see Kouvelis et al. 2013 for a detailed discussion), as opposed to our
inter-period utility.
Kouvelis et al. (2013) is the first paper to employ an inter-period utility for the joint inventory
control and financial hedging problem for a storable commodity in a multi-period setting. More
specifically, they consider a risk-averse buyer procuring a storable commodity from a spot market
and a fixed-price long-term contractual channel, and having access to the financial derivatives. In
each period, the buyer maximizes the mean-variance utility of the net present value (NPV) of the
total cash flow over the remaining periods, taking into account the future decisions under the same
criterion. However, it is notable that since the mean-variance utility is not time-consistent, the
future decisions under the same criterion may not be optimal for the objective function of current
period.
Our paper addresses the same issue of Kouvelis et al. (2013), but without the presence of longterm contract, and with the objective of maximizing the mean-variance utility of the terminal
wealth. We employ the dynamic programming approach to obtain a time-consistent optimal policy,
explicitly accounting for the fact that the mean-variance utility itself is not time-consistent. Our
7
analysis shows that the joint replenishment and hedging policy can be decomposed as follows: (1)
the optimal inventory policy is derived from solving a sequence of myopic optimization problems,
and thus myopic base-stock policies are optimal; and (2) the financial hedging portfolio is to offset
the price risk associated with the inventory carried over to the next period and the profit-to-go
as a function of future price. Our result confirms that the operational (inventory) decisions can
be separated from financial decisions. Differing from results in Chen et al. (2007), we show that
financial hedges do affect optimal inventory policies (myopic policies are no longer optimal without
financial (forward) hedges). In the last period, financial hedges increase inventory order-up-to level,
whereas in previous periods financial hedges can reduce inventory order-up-to level in comparison
to the unhedged solution. We further extend our analysis to account for backlogs, price-sensitive
demand, and transaction costs. In particular, in the presence of transaction costs, the myopic
inventory policy is no longer optimal, but the inventory decision is still separable from financial
hedging decisions when exotic hedges with forwards and options of all strikes may be used.
3.
3.1.
The Model
Problem Description
Consider a risk averse firm purchasing a storable commodity from a volatile spot market. This
commodity serves as a raw material which can be processed into an end product with uncertain
demand and fixed selling prices. Assume that there exists a spot market and an associated financial
market for financial derivative contracts written on the spot price. The inventory is reviewed
periodically and the periods are indexed by t = 0, 1, 2, · · · , T + 1. Period 0 is the beginning period
and period T + 1 is the ending period. The firm serves the demand occurring in period t = 1, · · · , T ,
denoted by dt , with a fixed unit selling price rt . Unmet demand is lost. The excess inventory is
carried over to the next period which incurs a holding cost ht for each unit of inventory.
We assume that this commodity product can be sold or bought from spot market with current
spot price at the beginning of each period. The replenishment leadtime is zero. Assume that the
commodity spot market is liquid in the sense that there is no transaction cost and the bid-ask
spread is zero. Let St be the risk-adjusted spot price at the beginning of period t. Denote by s the
realization of the spot price. The spot price process {St , t = 0, 1, · · · , T } is a Markov process.
Demand in each period may depend on the spot price. Without loss of generality, we assume
that the support of demand is R+ . Let Φt (·|s) and ϕt (·|s) be the distribution and density functions
of demand given a spot price s. Assume that Φt (ξ, s) is twice differentiable in (ξ, s). As Gaur and
Seshadri (2005), one may specify the demand as the form
dt = a + bs + ϵt ,
8
where ϵt is an error term independent of s such that Et [ϵt ] = 0 and E[ϵ2 ] < ∞.
We denote by Ft the filtration generated by (St , dt ). Throughout the paper, we make the no
arbitrage assumption and assume a risk-neutral probability measure Q (see, e.g., Geman and Ohana
2008). For convenience, let Et [·] = E[·|Ft ] = E[·|St = s] denote the conditional expectation operator
under Q.
According to the competitive storage theory, the commodity price satisfies the following noarbitrage arguments (see, e.g., Working 1948, Williams and Wright 1991, Routledge et al. 2000).
Assumption 1. The spot price and holding cost satisfy the no-arbitrage condition, i.e.,
s + ht − αE[St+1 |S(t) = s] ≥ 0,
where α is the risk-free discount factor.
The financial market is assumed to be complete. All the financial contracts (e.g., forwards/futures
and options) are written on spot price and fairly priced. For example, the forward prices are
determined by the expected spot price under a risk neutral probability measure at maturity of the
contract. Let ft,τ (s) be the forward price quoted in period t and for delivery in period τ ≥ t given
the spot price at t, s. In particular, ft,τ (s) = Et [Sτ ] and ft,t = St . The no arbitrage assumption
implies that the forward prices are martingale under P, i.e., Et [ft,τ ] = ft,t = St . Let Ht (St+1 ) be the
net profit (or marked-to-market payoff) of a portfolio under the hedging strategy in period t with
a random payoff depending on the realization of the spot price St+1 in period t + 1. Without loss
of generality, we let function Ht represent a hedging strategy. Let Ht be the set of feasible hedging
strategies. We impose the following assumption on the payoff function of the heding portfolio.
Assumption 2. For all t and for any Ht ∈ Ht , the expected profit from the hedging strategy is
zero, i.e., Et [Ht (St+1 )] = 0.
This assumption implies that the financial derivatives are fairly priced and thus the expected profit
earned from the hedging portfolio is zero. This is a common assumption in finance literature (see,
e.g., Brown and Toft 2002).
3.2.
Dynamic Programming Formulation
Note that in period 0 only hedging decision is made. From period 1 to T , a joint inventory and
hedging decision is made. Let xt be the inventory level at the beginning of period t and yt be the
order-up-to level in period t such that yt ∈ R+ . The sequence of events in each period t > 0 is as
following: (1) observe the spot price St = s and obtain the payoff of last period’s financial hedge
Ht−1 (St ); (2) place an order yt − xt (bid if yt > xt or ask if yt < xt ) from spot market, and choose
9
a hedging portfolio Ht which matures at the next period; (3) demand dt occurs; and (4) holding
costs are incurred for ending inventory of the level (yt − dt )+ .
Let U be the set of admissible joint replenishment and financial hedging policies. Under any
policy u ∈ U , the dynamics of inventory level are
xt+1 = (yt − dt )+ , t = 1, · · · , T.
The operational profit function of each period is denoted by
π̃t (xt , yt , St ) = rt min(yt , dt ) − St (yt − xt ) − ht (yt − dt )+ , t = 1, 2, · · · , T,
π̃T +1 (xT +1 , ST +1 ) = xT +1 ST +1 .
Note that π̃0 = 0. For simplicity, we assume that x1 = x0 = 0.
Let W̃t be the initial wealth at the beginning of period t. Under a joint inventory and hedging
strategy, the dynamics of wealth are represented by
W̃t+1 = α−1 [W̃t + π̃t (xt , yt , St )] + Ht (St+1 ), t = 1, · · · , T
and the terminal wealth is expressed as
W̃T +1 = α−1 [W̃T + π̃T (xT , yT , ST )] + ST +1 xT +1 + HT (ST +1 )
T
∑
−(T +1)
=α
W̃0 +
α−(T +1−t) [π̃t (xt , yt , St ) + Ht−1 (St )] + ST +1 xT +1 + HT (ST +1 ),
t=1
where W0 is the initial wealth level which is an exogenously given positive constant. For convenience,
we redefine the operational profit associated with period t’s operation as
πt (yt , St , St+1 ) = rt min(yt , dt ) − St yt − ht (y − dt )+ + αSt+1 (yt − dt )+ ,
= (rt − St )yt − (rt + ht − αSt+1 )(yt − dt )+ ,
with π0 = 0 and πT +1 = 0. This profit function counts the total profit from the sales revenue less the
holding cost and plus the clear value of the ending inventory at the beginning of the next period.
Since the market is liquid and there is no backlog, the excess inventory can be sold back to the
market at the spot price of the next period without any transaction cost, and thus the inventory
replenishment decision of the next period does not depend on how much inventory is on hand. This
treatment effectively allows us to separate the cash flow related to the current period’s inventory
decision from the cash flow related to the next period’s inventory decision.
Redefine the dynamics of wealth process as
Wt+1 = α−1 [Wt + πt (yt , St , St+1 ) + αHt (St+1 )], t = 0, 1, · · · , T.
(1)
10
The terminal wealth can be expressed as
WT +1 =
T
∑
α−(T +1−t) [πt (yt , St , St+1 ) + αHt (St+1 )].
t=0
Notice that WT +1 = W̃T +1 .
The firm’s objective is to maximize the mean-variance utility of the terminal wealth at the end of
the planning horizon. In particular, the dynamic joint replenishment and financial hedging problem
of the firm is given by
max{Ut := Et [WT +1 ] − λV art [WT +1 ]},
u∈U
(2)
subject to the wealth dynamics (1).
Remark 1 (Time Consistency). The variance risk measure is time-inconsistent while the
expectation operation is time-consistent. To see this, we apply the law-of-total-variance (e.g., Weiss
2005) such that
V art (WT +1 ) = Et [V art+1 (WT +1 )] + V art [Et+1 [WT +1 ]], t < T + 1.
Clearly, the period-(t + 1) variance is larger than the expected variance at period-(t + 1). Thus, the
hedging strategy in period t should account not only for the expected period-(t + 1) variance of the
terminal wealth, but also the variance of period-(t + 1) expected terminal wealth. Then, given any
admissible policy u, the mean-variance utility Ut satisfies the recursive representation
Ut = Et [Ut+1 ] − λV art [Et+1 [Wt+1 ]].
That is, the expected mean-variance utility in period t consists of the expected future utility and
an adjustment. This adjustment enables us to determine the optimal policy by backward induction.
This derived optimal policy is time-consistent in the sense that the firm optimally chooses the policy
taking into account that he or she will act optimally in the future. We refer to Basak and Chabakauri
(2010) for a more detailed discussion about the dynamic mean-variance analysis framework and
the time consistence of the derived optimal policy.
Let J˜t (Wt , s) as the mean-variance utility under the optimal policy u∗ which solves (2) given the
wealth level and spot price at the beginning of period t. The following proposition presents the
dynamic programming formulation for J˜t .
Proposition 1. J˜0 (W0 , s) satisfies the following optimality equation:
[
]
J˜0 (W0 , s) = E0 [J˜1 (W1 , S1 )] − λ min V ar0 α−T H0 (S1 ) + v1 (S1 ) .
H0 ∈H0
(3)
11
For t = 1, · · · , T , J˜t (Wt , s) satisfies the following optimality equation:
{
}
[ −(T +1−t)
]
˜
˜
Jt (Wt , s) = max
Et [Jt+1 (Wt+1 , St+1 )] − λV art α
[πt + αHt (St+1 )] + vt+1 (St+1 ) ,(4)
yt ≥0,Ht ∈Ht
where vt (s) is the expected profit-to-go function (accounted to period T + 1) under the optimal policy
and satisfies the following recursive equation:
vt (s) = Et [α−(T +1−t) πt (yt∗ (s), St , St+1 )] + Et [vt+1 (St+1 )],
(5)
with the terminal value vT +1 (s) = 0. Here, yt∗ (s) is the optimal order-up-to level in period t when
the spot price is s.
Equation (3) shows that in period 0 only the hedging decision is made. For t = 1, ..., T , the
optimality equation (4) states that the period-t mean-variance of terminal wealth under the optimal
policy equals the maximum of expected period-(t + 1) mean-variance of terminal wealth less the
product of λ and the variance of period-(t + 1) expected terminal wealth. The second term serves as
a time-consistent adjustment term which enables us to determine the optimal policy by backward
induction. The derived optimal policy is time-consistent in that the firm optimally chooses the
decisions taking into account that she will act optimally in the future periods.
The following proposition shows that the formulation (4) for the problem (2) can be transformed
into a simpler equivalent problem by separating the wealth at the beginning of each period from
the objective function.
Proposition 2. The optimal mean-variance utility function J˜t (Wt , s) can be decomposed into
two parts as J˜t (Wt , s) = Jt (s) + α−(T +1−t) Wt , where Jt (s) satisfies the following optimality equations:
J0 (s) = E0 [J1 (S1 )] − λα−2T min V ar0 [αT v1 (S1 ) + H0 (S1 )],
(6)
Jt (s) = max{α−(T +1−t) gt (y, s)} + Et [Jt+1 (St+1 )], t = 1, ..., T,
(7)
H0 ∈H0
y≥0
where JT +1 (s) = 0 and
[
−(T +1−t)
]
At (s)V ar[(y − dt ) ] + α ψt (y, s) ,
gt (y, s) = Et [πt ] − λα
{
}
+
T −t
ψt (y, s) = min V art [St+1 Et [(y − dt ) ] + α vt+1 (St+1 ) + Ht (St+1 )] .
+
2
Ht ∈Ht
(8)
(9)
where At (s) = Et [(rt + ht − αSt )2 ] = (rt + ht − αft (s))2 + α2 V art [St+1 ].
The equations (6) and (7) are equivalent to (3) and (4). Note that Jt (s) is the period-t expected
mean-variance of the terminal wealth with subtraction of the initial wealth in period t. The separation of the initial wealth in the objective function allows us to separate Jt+1 from the optimization
12
problem in each period and concentrate on the expected single period profit Et [πt ] and the timeconsistent adjustment term.
The optimality equation (7) implies that in each period t the optimal policy is independent of
the initial wealth of this period. The inventory and financial hedging decisions depend only on the
initial price and thus the current inventory decision will not affect next period’s decisions. Moreover,
the time-consistent term is divided into two parts: the first one is related to the variance of overage
inventory which is carried over to the next period, and the second one is about the variance of the
cash flows related to the spot price: the value of expected overage inventory carried to next period,
the period-(t + 1) expected profit-go-to, and the financial hedging term. This decomposition implies
that the financial hedges are used to mitigate the price risk (by minimizing the variance of the
cash flows related to the spot price of the next period), while the risk due to demand uncertainty
(i.e., the variance of overage inventory) cannot not be financially hedged. In other words, demand
risk is hedged with physical inventory and price risk can be hedged by financial instruments. In
general, the demand risk cannot be perfectly hedged.
The optimal policy can be solved by backward induction with (7). The optimality equation (7)
also indicates that the joint inventory and financial hedging decision can be solved by two sequential
steps. First, given any inventory decision y, the optimal hedging strategy is derived by solving the
problem (9) from which we can see that the financial hedges should be constructed to offset the
cash flow at the beginning of the next period, St+1 Et [(y − dt )+ ], and the discounted profit-to-go
αT −t vt+1 (St+1 ). Second, the optimal inventory decision is obtained by solving the problem in (7),
given the optimal inventory level dependent hedging strategy.
Before closing this section, we introduce the evolution of the variance. Let Vt (s) be the conditional
variance of the terminal wealth in period t, t = 0, 1, ..., T . The preceding analysis implies that Vt (s)
satisfies the following recursive equations:
V0 (s) = E0 [V1 (S1 )] + α−2T V ar0 [αT v1 (S1 ) + H0 (S1 )],
Vt (s) = Et [Vt+1 (St+1 )] + α−2(T +1−t) [At (s)V ar[(yt∗ − dt )+ ] + α2 ψt (yt∗ , s)], t = 1, ..., T.
4.
Optimal Policies
This section first addresses the optimal hedging and inventory control policy with forwards and
then develop the replication strategy when there exist options of all strikes.
4.1.
Hedging with Vanilla Derivatives
Vanilla derivatives refer to the standard derivative contracts such as futures, forward and simple
options. Futures and forward contracts are the most commonly traded derivative contracts in
commodity markets. The trading mechanisms of the futures and forwards are quite similar except
13
the fact that futures are traded on exchanges and forwards are traded in over-the-counter (OTC)
market. For the sake of simplicity, we ignore the difference in trading details between the futures
and forwards.
The following theorem characterizes the structure of the integrated inventory replenishment and
hedging policies with vanilla derivatives.
Theorem 1 (Forward Hedge). Assume that Ht contains at least a forward contract maturing
at next period. For all t,
(a) a myopic base-stock policy is optimal and the optimal base-stock level is obtained by solving
the following sub-problem:
{
}
max Et [πt ] − λα−(T +1−t) At (s)V ar[(y − dt )+ ] , t = 1, ..., T.
y≥0
(10)
In particular, if rt ≤ s, the least optimal inventory level yt∗ (s) = 0; otherwise, yt∗ (s) is the unique
solution of the following equation:
[
]
rt + ht − αft (s) − 2α−(T +1−t) λE[(y − dt )+ ]At (s) Φ̄t (y |s) − (s + ht − αft (s)) = 0.
(11)
(b) the optimal hedging portfolio consists of a position of shorting E[(yt∗ (s) − dt )+ ] units of
forwards maturing at next period and a portfolio H̃t∗ which is the solution of the following subproblem:
min V art [H̃t (St+1 ) + αT −t vt+1 (St+1 )]
H̃t ∈Ht
(12)
(c) in particular, if only forward contracts maturing at next period are available, then
H̃t∗ = −αT −t
and
ψt (y, s) = α
2(T −t)
Covt [St+1 , vt+1 (St+1 )]
[St+1 − ft (s)],
V art [St+1 ]
[
]
(Covt [St+1 , vt+1 (St+1 )])2
V art [vt+1 (St+1 )] −
.
V art [St+1 ]
Theorem 1 characterizes the optimal policy in the presence of forward contracts. Specifically,
part (a) shows that a myopic base-stock policy is optimal. Part (b) shows that the optimal hedging
position of the forward contracts consists of two parts: Short E[(yt∗ (s) − dt )+ ] units of forward and
long −αT −t
Covt [St+1 ,vt+1 (St+1 )]
V art [St+1 ]
(short if it is negative) units of forwards that mature next period.
The first part offsets the variation due to the excess inventory carried over to next period and the
second part is to hedge against the risk associated with the future cash flow, vt+1 (St+1 ). The final
hedging position is the combination of the two parts:
(
)
∗
∗
+
T −t Covt [St+1 , vt+1 (St+1 )]
Ht = − E[(yt (s) − dt ) ] + α
[St+1 − ft (s)].
V art [St+1 ]
14
It is a long (short) position if E[(yt∗ (s) − dt )+ ] + αT −t
Covt [St+1 ,vt+1 (St+1 )]
V art [St+1 ]
is negative (positive).
This theorem implies that, in the presence of the forward contract maturing at the next period,
one could make the inventory decision independently, knowing that the risk associated with the
expected excess inventory will be eliminated by a forward hedge. Then Jt can be rewritten as
Jt (s) = α−(T +1−t) ḡt (s) − λα−2(T −t) ψ̄t (s) + Et [Jt+1 (St+1 )],
{
}
ḡt (s) = max Et [πt ] − λα−(T +1−t) At (s)V art [(y − dt )+ ] ,
y≥0
[
]
ψ̄t (s) = min V art αT −t vt+1 (St+1 ) + H̃t (St+1 ) .
H̃t ∈Ht
(13)
(14)
That is, the joint optimization problem (7)-(9) can be decomposed into two sub-problems: One
only involves inventory decisions and the other only involves hedging decisions. This decomposition
allows us to compute the original dynamic program in four steps: (1) Derive the optimal myopic
base-stock level for each period by solving the inventory-related sub-problem (13); (2) Compute
the expected profit-to-go function vt for all the periods; (3) Compute the optimal portfolio H̃t∗
of (14) using the vt ; and (4) Obtain the optimal hedging portfolio by combing the short position
related to the expected excess inventory and H̃t∗ .
Note that although we restrict our attention to the forward contracts, it is straightforward to
observe from (14) that this decomposition applies to the general setting where the hedging portfolio
contains not just forwards but also other derivatives contracts (e.g., options).
Remark 2. The optimality of the myopic base-stock policy relies on the existence of the forwards (maturing in next period). Without the forward contract maturing at next period, the risk
associated with overage inventory cannot be perfectly replicated and therefore the determinant of
the optimal base-stock level yt∗ (s) involves the profit-to-go vt+1 , which implies that the optimal
base-stock policy may not be myopic any more and the joint optimization problem may not be
decomposed into two separate sub-problems.
This decomposition has an important practical implication. In practice, inventory decisions are
made by operations managers while the financial hedging decisions are made by financial managers. Our result suggests that the management can allow the operations manager to manage the
inventory decision separately and myopically. But, when making the hedging decisions, the risk
manager must take into account the optimal inventory policy to forecast the resulting future cash
flow. Therefore, the financial hedges simplify the operations manager’s task and help clarify the
relationship between operations management and financial risk management within an organization.
The following corollary is a direct application of Theorem 1.
15
Corollary 1. Suppose the demand is perfectly correlated to the spot price and has the specification dt = at + bt s for constants at , bt such that dt ≥ 0. Then, for t = 1, · · · , T ,
(a) yt∗ (s) = a + bs if rt > s and yt∗ (s) = 0 if rt ≤ s.
∑T
(b) vt (s) = Et [ i=t αi−t (rt − Si )+ (ai + bi Si )].
(c) If only the forwards contracts maturing at next period are available, then the optimal hedging
portfolio is
Ht∗ = −
Covt [St+1 , αT −t vt+1 (St+1 )]
[St+1 − ft (s)].
V art [St+1 ]
(d) If options contracts maturing at any future periods are available and b = 0, i.e., dt = a, then
the optimal hedging strategies is to short ai units of put options with strike price rt maturing at
future periods i = t + 1, ..., T .
Corollary 1 shows that when the demand is perfectly correlated to the spot price, i.e., only price
risk exists, the ordering quantity of each period is simply to order the demanded quantity. But
vt (s) is still nonlinear in s (due to the term (rt − St )+ ), which implies that the price risk cannot be
perfectly hedged by forward contracts. For deterministic demand, the price risk can be perfectly
hedged by entering the short positions in put options.
We next examine the impact of risk-aversion on the inventory policy. Let yt∗ (s; λ) be the optimal
inventory decision with respect to λ. The following corollary shows the effect of risk aversion on
the optimal inventory decision.
Corollary 2. For all t, yt∗ (s; λ) is decreasing in λ.
This result shows that the more risk averse the decision maker is the less inventory the firm
orders, and thus the fewer forwards to short. In particular, as λ = 0, the yt∗ (s; λ) reduces to a
newsvendor solution (if rt > s):
yt∗ = Φ̄−1
t ((s + ht − αft (s))/(rt + ht − αft (s))).
Let πt∗ , Ut∗ , Et [WT∗+1 ] and V art [WT∗+1 ] be the period-t expected operational profit, mean-variance,
the expected terminal wealth and variance of the terminal wealth under the optimal policy. We
have the following intuitive result.
Corollary 3. For all t, E[πt∗ ], Ut∗ , Et [WT∗+1 ] and V art [WT∗+1 ] are all decreasing in λ.
16
4.2.
Hedging with Continuum of Option Strikes
In financial markets, since option contracts traded in exchanges normally have limited number of
strikes, exotic derivatives are structured to develop more sophisticated hedging or trading strategies.
In financial literature, it is common to assume that there exists a continuum of options of all
strikes as a proxy to a frictionless financial market. As argued by Carr and Madan (2001), the
assumption of a continuum of strikes is essentially the counterpart of the standard assumption of
continuous trading, which serves as a reasonable approximation to a market environment where
there is a large but finite number of European options strikes (e.g., options for S&P 500). In a
single-period hedging model, they show that a twice-continuously differentiable payoff function can
be replicated by a portfolio that consists of a spectrum of European put and call options with all
strike prices, forwards and risk-less bonds. This type of derivative contracts belong to the exotic
derivative contracts. In the following analysis, we will follow Car and Madan’s (2001) approach to
construct the optimal exotic hedging strategies.
Note that Car and Madan’s (2001) approach requires the payoff function to be twice differentiable. To employ their approach, we assume that the spot price satisfies the following condition.
Assumption 3. For all t and any real function v(·) which is twice differentiable almost everywhere, the conditional expectation E[v(St+1 )|St = s] is twice differentiable in s almost everywhere.
For example, for the geometric Ornstein-Uhlenbeck process (also called Schwartz’s One-Factor
model, Schwartz 1997):
d ln S = κ(α∗ − ln S)dt + σdz ∗ (t)
(15)
where κ is the mean-reverting rate, α∗ = η − σ 2 /2 − γ is the risk adjusted long-run mean (η is
the long-run mean and γ is the market price of risk) and z ∗ is a Brownian motion under the risk
neutral measure. Assume the length of each period is equal to ∆. It follows from Schwartz (1997)
that the discrete-time dynamics of St+1 are of the form
( (
)
)
St+1 = exp α∗ 1 − e−κ∆ + e−κ∆ ln St + ϵt+1 ,
)
( √
1−e−2κ∆
σ .
where ϵt ∼ N 0,
2κ
(16)
′′
Apparently, for any twice differentiable function v with finite E[v ′ (S)] and E[v (S)],
v(St+1 ) is twice differentiable in St and then the conditional expectation E[v(St+1 )|St = s] =
[ ( −κ∆
)]
E v se
exp (α∗ (1 − e−κ∆ ) + e−κ∆ + ϵt+1 ) is twice differentiable in s.
We now proceed to present our key result of this section.
Theorem 2 (Exotic Hedge). Suppose that there exist forwards and European calls and puts
of all strikes, and Assumption 3 holds. For all t,
17
(a) the myopic base-stock policy is optimal. If rt ≤ s, the least optimal base-stock level yt∗ (s) = 0;
otherwise, yt∗ (s) is the unique solution of the following FOC (first-order-condition):
[
]
rt + ht − αft (s) − 2α−(T +1−t) λE[(y − dt )+ ]At (s) Φ̄t (y |s) − (s + ht − αft (s)) = 0.
(17)
′
(b) the optimal hedging portfolio is to short Et [(yt∗ (s) − dt )+ ] + αT −t vt+1 (s) units of forwards,
′′
′′
αT −t vt+1 (K)dK puts for all strikes K < s and αT −t vt+1 (K)dK calls for all strikes K > s maturing
at next period. That is,
[
′
∗
Ht (St+1 ) = − (Et [(yt∗ (s) − dt )+ ] + αT −t vt+1 (s))[St+1 − ft (s)]
]
∫ s
∫ ∞
T −t ′′
+
T −t ′′
+
+
α vt+1 (K)[(K − St+1 ) − Pt (K)]dK +
α vt+1 (K)[(St+1 − K) − Ct (K)]dK ,
0
s
where Pt (K) = Et (K − St+1 )+ and Ct (K) = Et (St+1 − K)+ .
Theorem 2 characterizes the structure of the optimal joint inventory control and hedging strategies with complete term structure. Part (a) shows that the optimal ordering policy is described
by a myopic state-dependent base-stock level, yt∗ (s). In any period t = 1, · · · , T , if rt > s, then
order up/sell down to yt∗ (s); otherwise, no order is placed. The ordering policy is myopic since the
optimal base stock level in a period does not depend on the decisions to make in the future. This
significantly simplifies the computation of the optimal solution.
Part (b) describes the structure of the optimal hedging strategy. The hedging portfolio consists
of two parts that are associated with the inventory decision of current period and the profit-togo function of the next period, respectively. First, the optimal hedging strategy requires shorting
Et [(yt∗ (s) − dt )+ ] units of forwards that mature next period. The intuition behind this is as follows.
In the first part, given the order-up-to inventory level, yt∗ (s), the expected inventory carried over to
next period is Et [(yt∗ (s) − dt )+ ]. Thus, the associated risky payoff is Et [(yt∗ (s) − dt )+ ]αSt+1 ] with the
variance being (Et [(yt∗ (s) − dt )+ ])2 V art (St+1 ). Shorting Et [(yt∗ (s) − dt )+ ] units of forwards offsets
the variation caused by the excess inventory carried over to next period.
The second part of the hedging strategy is related to vt+1 (St+1 ), the profit-to-go function which
represents the expected total profit from next period. We follow Carr and Madan (2001) to decompose the profit-to-go function into three components which can be interpreted by the portfolio of
′
(s)
the forwards and options. Specifically, the optimal hedging decision requires to short αT −t vt+1
′′
units of forward (in addition to those related to inventory decision), αT −t vt+1 (K)dK puts for all
′′
strikes K < s and αT −t vt+1 (K)dK calls for all strikes K > s maturing at next period. As a result,
under the optimal hedging strategy, the next-period expected profit vt+1 (St+1 ) is completely offset
by the proposed portfolio.
18
Now, under the optimal hedging strategies, the equation (8) is equal to
gt (y, s) = Et [πt ] − λα−(T +1−t) At (s)V ar[(y − dt )+ ],
where the second term involves the variance associated with dt . This implies that even with complete term structure of the financial instruments, we are not able to completely hedge all the
risk. In fact, we can only mitigate risks related to market spot price (i.e., St+1 Et [(yt∗ (s) − dt )+ ]
and vt+1 (St+1 )) with the financial instruments. The risk associated with the demand is hedged
with physical inventory. This demonstrates the different roles of inventory and financial hedge in
managing the demand and market price risks.
5.
Inventory Hedges versus Financial Hedges
This section discusses the setting without the financial hedges, namely inventory hedge. Without
the financial hedges, for t = 1, ..., T , the optimality equation becomes
Jt (s) = max {α−(T +1−t) ĝt (y, s)} + Et [Jt+1 (St+1 )],
y≥0,z≥0
(18)
where
[
ĝt (y, s) = Et [πt ] − λα
−(T +1−t)
]
At (s)V ar[(y − dt ) ] + α ψt (y, s) ,
+
2
ψt (y, s) = V art [St+1 Et [(y − dt )+ ] + αT −t vt+1 (St+1 )].
(19)
(20)
Let ŷt∗ be the optimal inventory decision without the financial hedges.
We first analyze the last period’s problem.
Proposition 3. In period T , without financial hedges,
(a) if rt ≤ s, the optimal inventory decision ŷT∗ (s) = 0; otherwise, the optimal inventory decision
ŷT∗ (s) satisfies the following first-order condition:
[
]
0 = rt + hT − αfT (s) − 2α−1 λE[(y − dT )+ ]AT (s) Φ̄T (y, s) − (s + hT − αfT (s))
−2λαV arT [ST +1 ]ΦT (y, s)ET [y − dT ]+ .
(21)
(b) in particular, when rt > s, ŷT∗ (s) < yT∗ (s), i.e., the unhedged inventory level is lower than
hedged inventory decision.
Part (a) characterizes the optimal inventory solution. Part (b) states that the financial hedges lead
to a higher inventory level compared to the system without financial hedges and therefore provides
a better service level. This result is consistent with the numerical findings in Kouvelis et al. (2013)
that the financial hedges increase customer service level by increasing the inventory level. In other
19
words, the inventory hedges are complementary to the financial hedges. Similar insights are also
found in newsvendor models (see, e.g., Gaur and Seshadri 2005 and Gao et al. 2011).
Although the relationship between inventory and financial hedges in the last period are essentially
the same as that in the newsvendor model, it may be different in the previous periods. The following
proposition gives a condition that the hedged inventory decision is less than the unhedged.
Proposition 4. For any period t = 1, · · · , T − 1, suppose that yt∗ (s) > 0. Then, ŷt∗ (s) > yt∗ (s) if
and only if
E[(yt∗ (s) − dt )+ ] < −αT −t
Covt (St+1 , vt+1 (St+1 ))
.
V art (St+1 )
This proposition shows that the optimal unhedged inventory decision may not necessarily be less
than the optimal inventory decision under financial hedges when the spot price is negatively correlated to the expected next-period profit-to-go (as a function of the next period’s price). Intuitively,
the expected profit is a decreasing function of the purchasing price and thus the profit-to-go is likely
to be negatively correlated to the spot price. To justify this intuition, we consider the following
special case.
Proposition 5. If the demand is perfectly negatively correlated with the spot price with the form
s+ht −αft
dt = a − bs, a > 0, b > 0, then ŷt∗ > dt if and only if λ < − 2αCovt (S
.
t+1 ,vt+1 (St+1 ))
∑T
Note that vt (s) = Et [ i=t αi−t (rt − Si )+ (ai − bi Si )] which is a strictly decreasing function of St = s.
Then it is clear that St+1 and vt+1 (St+1 ) are negatively correlated, i.e., Covt (St+1 , vt+1 (St+1 )) < 0.
This proposition shows that, when the demand is perfectly negatively correlated with spot price
and the risk-averse degree is sufficiently high, the unhedged strategy orders more inventory in
period t to hedge the future inventory risk. Recall that, in the presence of financial hedges, the
optimal inventory decision is to order dt .
The discussion above sheds new light into the relationship between physical inventory and financial hedges: Financial hedges can be a substitute of physical inventory in a multi-period inventory
system, whereas it is a complement to the inventory in a single-period inventory system where
financial hedges lead to a higher inventory and thus a higher service level. This is the key difference
between the storable commodities (e.g., oil and gas) and perishable commodities (e.g., electricity). The implication gained from the newsvendor model may no longer apply to the dynamic
model of storable commodity inventories though in conventional inventory models many results
found in newsvendor model generally hold in the corresponding dynamic models. This justifies the
importance of explicitly addressing the dynamic hedging and inventory control problem.
20
6.
How Financial Hedges Create Value
A conventional thought is that hedging under no arbitrage condition does not increase or decrease
profit. Recall that for a fixed λ, exotic hedges have the same expected terminal wealth and inventory
level as that of forward hedges, though the variance of exotic hedges is smaller.
For convenience, we use use superscripts e, f and u to indicate these strategies (or market
profiles): hedging with forwards and options with all strikes (exotic hedges), hedging with forwards
only (forward hedges) and unhedged inventory policy. The following lemma compares the meanvariance utilities, expected terminal wealth and variances under difference hedging strategies for
the same λ. The proof is straightforward and is therefore omitted.
Lemma 1. For any given λ, wealth and price levels at time t, Ut∗e ≥ Ut∗f ≥ Ut∗u , but Et [WT∗e+1 ] =
Et [WT∗f+1 ] and V art [WT∗e+1 ] ≤ V art [WT∗f+1 ].
Note that the risk-averse parameter λ reflects the risk tolerance level of a decision maker. A higher
risk tolerance implies a smaller λ. In practice, instead of giving λ directly, one normally specifies
the maximum variance level that he/she can tolerate while maximizing the expected return, i.e.,

 maxω∈U E0 WTω+1 ,
subject to
 V ar [W ω ] ≤ δ
0
T +1
where ω is a policy and U is the set of feasible policies, and δ > 0 is a given variance level. Let λ
be the Lagrange multiplier. Then the problem becomes
min max{E0 WTω+1 − λ[V ar0 [WTω+1 ] − δ]}.
λ
ω
(22)
For any i ∈ {e, f, u}, let λ∗i be the corresponding optimal Lagrange multiplier derived from the
Lagrange problem (22)
Let δ0 be the variance of the terminal wealth under the risk-neutral solution and WT0+1 the
expected risk-neutral terminal wealth. The following theorem characterizes the efficient frontier of
the three hedging strategies.
Theorem 3. Given the same wealth level and spot price at the beginning of the planning horizon,
and a tolerance level δ < δ0 , the following assertions hold.
(a) Exotic hedges dominate forward hedges which then dominate unhedged inventory policies,
i.e.,
U0∗e ≥ U0∗f ≥ U0∗u .
(b) For any i ∈ {e, f, u}, λ∗i > 0 and V ar0 [WT∗i+1 ] = δ but
E[WT∗e+1 ] ≥ E[WT∗f+1 ] ≥ E[WT∗u+1 ].
21
(c) λ∗i is decreasing in δ and then E[WT∗i+1 ] is increasing in δ. In particular, as δ → δ0 , E[WT∗i+1 ]
converges to risk-neutral expected terminal wealth WT0+1 .
(d) λ∗e ≤ λ∗f and thus yt∗e ≤ yt∗f .
Theorem 3 shows that the efficient frontiers are increasing and converge to the expected riskneutral terminal wealth level (see parts (a) and (c)). The efficient frontier under exotic hedge
dominates that under forward hedge and the efficient frontier under forward hedge dominates that
of unhedged inventory policies. Part (b) and (d) imply that for any given tolerance level, the exotic
hedges provide a higher expected terminal wealth level than forward hedges do; and the forward
hedges in turn provide a higher expected terminal wealth level than unhedged inventory policies
do. This indicates that financial hedging is profitable. Finally, part (d) shows that the complete
hedges yield a higher inventory level (service level) than the forward hedge.
These results have important practical implications. First, the risk-aversion parameter λ is rarely
specified directly in practice. Instead, it is natural to identify the risk-tolerance level δ, which
then determines the selection of λ as a Lagrange multiplier to penalize the variance. From this
perspective, one can argue that the risk-aversion parameter can be endogenously selected.
Second, for any given risk-tolerance level, exotic hedges dominate the vanilla (forward or simple
option) hedges and the optimal selection of the Lagrange multiplier binds the variance constraint,
which implies that the expected terminal wealth level under exotic hedges is higher than that under
a less complete hedge. Third, exotic hedges yield a higher inventory level than forward hedges do,
which also explains why hedging can be profitable. Fourth, exotic hedges have a lower risk-aversion
parameter λ than forward hedges do, which implies that compared to forward hedges, exotic hedges
allow one to be less sensitive to the risk and therefore to be able to gain more profit.
7.
Numerical Example
This section presents a numerical example to demonstrate and compare the structures of the
optimal integrated financial hedging and replenishment policies and their mean and variance performances under different hedging strategies: (1) forward hedges, (2) exotic hedges, and (3) no
(financial) hedge.
Specifically, we consider a two-period model, i.e., T = 2, with the length of each period being
∆ = 1. Assume that the risk-adjusted spot price process follows the Geometric Mean-Reverting
process (16). We use the calibration of Schwartz (1997) for crude oil price to specify the parameters
of the price process:
κ = 0.428, µ = 2.991, σ = 0.257, γ = 0.002.
The demand is expressed as dt = µd + ϵt where µd is the mean of the demand and the error terms
ϵt are i.i.d. normally distributed with mean 0 and variance σd2 . This implies that the demand is
22
normally distributed with mean µd and variance σd2 . Set µd = 10 and σd = 5. The initial wealth level
is assumed to be zero. The other cost and model parameters are rt = 50, ht = 5, α = 0.9 and λ = 0.01.
The optimal policy is computed by applying the value iteration approach on the corresponding
optimality equations. The results are demonstrated in figures (1)-(3).
Figures 1 demonstrates the optimal order-up-to levels with or without financial hedges. Recall
that as long as the forward contracts are used, the risk associated with the inventory carried over
to the next period is perfectly hedged and the firm solves the same optimization problem regardless
what other derivative contracts are used. Thus, the inventory decisions under forward hedges and
exotic hedges are identical. When comparing the hedged and unhedged inventory decisions, one
can see the relationship between the hedged and unhedged inventory order-up-to levels changes
when moving from the last period to the previous periods: In the last period t = 2, the hedged
order-up-to level is greater than the unhedged order-up-to level whereas in period 1 the hedged is
smaller than the unhedged. In other words, financial hedge increases the order-up-to level in the
last period but decreases the order-up-to level in the first period. Recall that newsvendor models
predict that financial hedges induce the firm to order more and thus financial hedges and inventory
are complements to each other (see, e.g., Gaur and Seshadri 2005). However, in a dynamic system
like ours, such a prediction is no longer true. In fact the opposite relationship is more likely to hold
in periods before the last period. This can be explained by the relationship between the profit-to-go
function vt (s) and the spot price s. As shown in Figure 3, the profit-to-go functions are decreasing
in the spot price, which implies that in each period t < T the next period’s price St+1 and the
profit-to-go vt+1 (St+1 ) tend to be negatively correlated. Since the negative correlation implies that
to offset the variation of the expected future cash flow it is optimal to enter a long position in
forward (or to buy forward contracts) if forwards are used, when there is no financial hedge, the
inventory carried over to the next period can play the role of forward hedges. That is, the financial
hedges and inventory are substitutes for each other.
Figure 2 demonstrates the hedging solutions. Note that the positive (negative) sign indicates a
long (short) position. We first discuss the solution under the forward hedge. In the last period, the
optimal hedging policy should offset the risk associated with the excess inventory, which implies a
short position. Thus the sign is negative. But in the previous period, the hedging positions are all
positive. This echoes previous discussion that the negative correlation between the spot price and
profit-to-go suggests that the forward hedger should enter a long position. More over, the forward
hedgers will buy (long) more in period 0 than in period 1. This can be explained by the fact that
the downward slop of v1 is greater than v2 (the change of v1 is greater the price increases from 1
to 40), i.e., the St+1 and Vt+1 are more negatively correlated in period 0. Under the exotic hedge,
we have the similar observations. Comparing the forward positions under the forward hedge and
23
optimal order−up−to level (t=2)
7
6.5
6.5
2
6
order−up−to level y*
order−up−to level y*1
optimal order−up−to level (t=1)
7
5.5
5
4.5
4
3.5
3
0
unhedged
hedged
10
6
5.5
5
4.5
unhedged
hedged
4
20
spot price s
30
Figure 1
3.5
0
40
10
20
spot price s
unhedged
futures hedge
perfect hedge
30
40
Ordering decisions
exotic hedge at the same period, one can observe that the forward positions under the forward
hedge are higher than that under the exotic hedge. This is can be explained by the fact that some
of the risks are hedged by the options.
Recall that only the period 0 and period 1 need the options. Different from the corresponding
forward positions, the period 0 has a lower position than period 1, implying that it is more likely
to enter a short position in period 0 than period 1. Note that the short position is due to the
convexity of the profit-to-go function (the second order derivative is positive). In general, compared
to the forward position, the number of options to long or short is relatively smaller, which can be
explained by the fact the profit-to-go functions are decreasing almost linearly.
T−t ’’
options hedging curve of strikes α
hedging positions of forwards
14
v (K)
0.8
forward hedge (t=0)
12
0.6
10
0.4
number of options
forward hedge (t=1)
8
6
exotic hedge (t=0)
4
2
0
−2
0
30
Figure 2
exotic hedge (t=0)
−0.4
forward hedge and exotic hedge (t=2)
20
spot price s
exotic hedge (t=1)
0
−0.2
exotic hedge (t=2)
10
0.2
40
−0.6
0
Hedging decisions
10
20
strike price K
30
40
24
4
x 10 variance of terminal wealth (t=0)
4
0
460
variance V
terminal wealth level v0
expected terminal wealth (t=0)
470
450
440
430
420
20
22
24
26
28
3
2
1
20
30
400
350
26
28
30
4
x 10 variance of terminal wealth (t=1)
2
1.5
300
20
22
24
26
28
1
20
30
expected terminal wealth (t=2)
22
24
26
28
30
variance of terminal wealth (t=2)
200
unhedged
hedged
unhedged
futures hedge
exotic hedge
2
6000
variance V
terminal wealth level v2
24
1
2.5
variance V
terminal wealth level v1
expected terminal wealth (t=1)
450
22
150
5000
4000
100
20
22
24
26
spot price s
Figure 3
28
30
20
22
24
26
spot price s
28
30
Means and variances of terminal wealth
Finally, we discuss the mean and variance performances under hedging or unhedged strategy.
As mentioned above, the left panels of Figure 3 show that the profit function is decreasing in the
spot price. The profits under the forward hedges and exotic hedges are identical and they are very
close to the profit without hedges. In the last period, the hedged profit is slightly higher than
the unhedged profit but the opposite is observed in the previous periods. This can be explained
as follows: The order-up-to level under financial hedges in the last period is higher (and thus the
profit is closer to the risk neutral one). But in the previous period the order-up-to levels under
financial hedges is lower, which leads to lower profits. Nevertheless, in this example, the difference
in profit is very small.
The right panels of Figure 3 show that in periods 0 and 1 the unhedged variances are greater
than the variances under forward hedges and the later are greater than that under exotic hedges. In
the last period, the forward hedge and exotic hedge have the same variance whereas the unhedged
25
variance is smaller than the hedged variance. This is due to the different optimal inventory decisions
under hedged and unhedged strategies.
In summary, this example demonstrates and compares the structural relationships between inventory control and financial hedging under different hedging strategies. The dynamic system does
behave differently from the static models such as the newsvendor model and the difference between
them sheds new light into the commodity inventory management.
8.
Extensions
This section extends the analysis for the basic lost-sales case to several important extensions: backorder, price-sensitive demand, and transaction costs. In these extensions, we focus on examining
the relationship between the inventory control and financial hedges.
8.1.
Backorders
In many B2B businesses, unmet demand is often backlogged due to the contractual obligation to
meet all the demand, which may incur backlog costs. Let bt be the unit backlog cost of period t.
Assume that each backorder occurs in a period must be met in the next period, which is subject
to the contractual arrangement. That is, if there is backlog at the beginning of a period, the firm
will order at least as much as the backlog to meet the backlog even when the spot price is high.
Then the inventory order-up-to level of each period can be any nonnegative value, which implies
that the profit-to-go function vt+1 only depends on spot price. The backordering model yields the
same Bellman equation as (7) except that the operational profit of period t, πt , is replaced by
πt (yt , St , St+1 ) = rt dt − St yt − ht (yt − dt )+ − bt (dt − y)+ + αSt+1 (yt − dt ),
(23)
Hence, the analysis of the lost-sales model applies directly to the backordering case and similar
insights can be obtained.
8.2.
Price-Sensitive Demand
In the preceding analysis, the demand is exogenously given. However, in many industries, especially
B2C businesses, customers may have certain degree of price sensitivity though the underlying
products are commodities or made of commodity materials (e.g., retail gasoline and heating oils,
food, etc.) . This section aims to extend the preceding analysis to the case of price-sensitive demand.
Specifically, we assume that the demand dt is of the additive form:
dt = µt (r, s) + ϵt ,
where µt (r, s) is a deterministic and strictly decreasing function of the price r and dependent on
the spot price, and ϵt is a random variable with zero mean. The distribution function of ϵt can be
26
dependent of the spot price of period t, St . The price level is chosen from an interval [r, r]. Then
the operational profit in period t is
πt (r, y, St , St+1 ) = (r − St )y − (r + ht − αSt+1 )(y − dt )+ .
The profit-to-go function vt depends only on the spot price.
Following the analysis of Section 3.2, we can obtain the Bellman equation: For t = 1, ..., T ,
Jt (s) =
max {α−(T +1−t) gt (r, y, s)} + Et [Jt+1 (St+1 )]},
(24)
y≥0,r∈[r,r]
where
−(T +1−t)
[
]
+
2
At (r, s)V art [(y − dt ) ] + α ψt (r, y, s) ,
gt (r, y, s) = E[πt ] − λα
{
}
+
T −t
ψt (r, y, s) = min V art [St+1 E[(y − dt ) ] + α vt+1 (St+1 ) + Ht (St+1 )] ,
Ht ∈Ht
(25)
where At (r, s) = Et [(r + ht − αSt )2 ].
Then, similar to the analysis of Section 4, the inventory and pricing decisions can be isolated as
long as a forward maturing at next period is in used (with a short position of E[(y − dt )+ ] for the
chosen price and inventory levels). Hence, the pricing and inventory decisions can be obtained by
simply solving the myopic joint pricing and inventory decision problems for all periods:
max {E[πt ] − λα−(T +1−t) At (r, s)V art [(y − dt )+ ]}.
y≥0,r∈[r,r]
Letting y = z + µt (r, s), the joint pricing and inventory decision optimization can be written as
{
}
+
−(T +1−t)
+
max max{(r − s)(zt + µt (r, s)) − (r + ht − αSt+1 )(z − ϵt ) − λα
At (r, s)V art [(z − ϵt ) ]} .
r∈[r,r]
y≥0
Clearly, for any fixed r, the objective function is unimodal in y and thus the optimal ordering
solution satisfies the first order condition. The optimal price can be searched linearly.
Using the optimal pricing and inventory decisions, the hedger can estimate the expected future
cash flows and obtain the profit-to-go function vt . Finally, the the optimal hedging decision can be
obtained by solving the problem (25).
Thus, the insight obtained in Section 4 can be strengthened by incorporating the pricing decisions: In the presence of financial hedges (in particular, the forward contract maturing at next
period), both the pricing and inventory decisions can be obtained myopically, which implies that
the marketing (pricing) and operational (inventory) decisions can be made independently. But the
financial hedging decision must take into account of the inventory and pricing decisions which
influence the prediction of future cash flows.
27
8.3.
Transaction Costs
Trading in the spot market often incurs transaction costs for both buying and selling activities. The
transaction costs include the market cost (bid/ask spread) and operational cost (e.g., transportation
cost). Our analysis can be extended to include all relevant variable transaction costs.
Let kb and ks be the unit transaction costs for purchases or sales in the spot market. In the
presence of the transaction costs, it is no longer costless to sell all the on-hand inventory to the
market and then buy up to the optimal inventory level. Then, at the beginning of each period,
the on-hand inventory does influence the optimal inventory decision. But, for consistency, we still
represent the operational profit of period t in a similar way of Section 3.2:
πt (xt , yt , St , St+1 ) = (rt − St )yt − (rt + ht − αSt+1 )(yt − dt )+ − kb (yt − xt )+ − ks (xt − yt )+ .
Then the Bellman equation is
Jt (x, s)
{
= max Et [Jt+1 (y − dt , St+1 )] + α−(T +1−t) Et [πt ]
y≥0
−α
−2(T +1−t)
λEt [V ardt [πt + α
T +1−t
}
vt+1 ((y − dt ) , St+1 )|St+1 ]]
+
(26)
− α−2(T +1−t) λ min V art [αSt+1 Edt [(y − dt )+ ] + αT −t Edt [vt+1 ((y − dt )+ , St+1 )] + Ht (St+1 )],(27)
Ht ∈Ht
and the profit-to-go, vt (x, s), has the following dynamics
vt (x, s) = Et [α−(T +1−t) ] + Et [vt+1 (yt∗ − dt , St+1 )]
where the the terminal profit function is
vT +1 (x, s) = x(s − ks )+ − xs = −x min(s, ks ).
In the objective of the optimal hedging decision problem, since the profit-to-go is also a function
of y, the risk associated with the excessive inventory Edt [(y − dt )+ ] cannot be perfected hedged by
simply entering a short position in forwards. To isolate the inventory decision problem from the joint
optimization, we need sophisticated derivative portfolio. Under similar conditions to subsection
4.2, we can use forwards and options of all strikes (exotic derivatives) to replicate the expected
profit-to-go Edt [vt+1 ((y − dt )+ , St+1 )] as a function of y and St+1 . Note that the expectation is taken
on the demand dt . Then, the optimal inventory decision can be obtained by solving the following
Bellman equation:
{
Jt (x, s) = max Et [Jt+1 (y − dt , St+1 )] + α−(T +1−t) Et [πt ]
y≥0
−α
−2(T +1−t)
λEt [V ardt [πt + α
T +1−t
}
vt+1 ((y − dt ) , St+1 )|St+1 ]] .
+
28
Thus, although the myopic inventory policy is no longer optimal when there are transaction costs,
the inventory decision can still be isolated from the financial hedging decision. With the optimal
inventory policies derived from the above dynamic program, one can then evaluate and replicate
the profit-to-go as a function of price and the current inventory decision.
The above discussion implies that if there are sufficient derivative instruments in the market
then the operations manager can make inventory decisions separately whereas the risk manager
must use the inventory policy to predict the future cash flows. This concurs with insight previously
obtained.
9.
Concluding Remarks
This paper addresses the joint inventory and financial hedging decision problem for a commodity
inventory system with lost sales. We provide a novel dynamic programming formulation which
allows us to derive a time-consistent optimal policy. We characterize the structure of the optimal
policies under both forward hedges and exotic hedges. We show that as long as the forward contract
is used the inventory decisions can be separated from the financial hedging decisions and myopic
base-stock policies are optimal. However, to derive the optimal hedging decision, one needs to
evaluate the profit-to-go based on the optimal (state-dependent) base-stock levels in the future.
When there exist forwards and options of all strikes, a hedging portfolio can be constructed to
replicate the profit-to-go, which allows us to separate the inventory and financial decision in a very
general context. We also show that different from those newsvendor models in literature, financial
hedges may lead to lower order-up-to level. We further show that these results hold in rather
general frameworks (e.g., backlog, price-dependent demand and transaction cost). These results
provide new insights into the coordinated risk and operational management strategies.
Firstly, operational (inventory) decisions and financial hedging decisions interplay with each
other. On the one hand, financial hedges allow us to decompose a dynamic inventory decision
problem into a sequence of myopic decision problems, which significantly simplifies the inventory
decision process. On the other hand, financial hedging decisions rely on inventory strategies, since
the future cash flow depends on the inventory policies. Although some of these points have been
shown by Kouvelis et al. (2013), our results show that they hold in more general frameworks even
when myopic inventory policies may no longer be optimal (e.g., when there are transaction costs).
In real business world, the operational decisions and financial decisions are often made separately,
partially because of the difficulty in coordinating the decisions between operational and financial
managers. Our model suggests that the operational managers can still enjoy their independence
(of the financial decisions) while financial managers must be able to understand the cash flow
implications of operational strategies.
29
Secondly, financial hedges could be substitutes to inventory. In the single-period newsvendor
model, or the last period of the dynamic model, the financial hedges and inventory levels are
complementary. However, in the previous periods of the dynamic model, financial hedges may lead
to lower inventory levels, which implies that the financial hedges could be a substitute for inventory.
In practice, a firm tends to order more inventory in anticipation of a higher price in the future.
Our result suggests that financial derivatives are better instruments to hedge the future price risk,
while the inventory decisions focus on the current demand (quantity) risk which cannot be hedged
by financial instruments.
Thirdly, financial hedging does create values. We show that given a risk constraint, financial
hedges lead to a higher terminal wealth. This is also quite intuitive: Financial hedging reduces the
risk, which allows the firm to focus more on its value maximization decisions.
In summary, integrating operational and financial risk management strategies is the best way to
address comodity risk in an uncertainty demand environment. Our analysis framework provides us
a useful conceptual tool to explore these insights.
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Appendix. Proofs of Statements
Proof of Proposition 1
By induction, we assume that the optimality equation (4) holds for t + 1, · · · , T . Let πi∗ denote the profit
function in period i under the optimal policy, i = t + 1, · · · , T . Taking expectation on WT +1 in period t yields
Et [WT +1 ] = α−(T +1−t) Wt + vt (s)
= Et [α−(T +1−t) (Wt + πt (yt , St , St+1 ) + αHt (St+1 )) + vt+1 (St+1 )]
= Et [Et+1 [α−(T −t) Wt+1 + vt+1 (St+1 )]]
= Et [Et+1 [WT +1 ]],
where
vt+1 (St+1 ) = Et
[ ∑
T
i=t+1
]
α−(T +1−i) [πi∗ (yi∗ (Si ), Si , Si+1 )] , t = 0, 1, · · · , T.
32
It is not hard to see that vt (s) satisfies the recursive equation (5). Therefore, given the initial wealth level
in period t, the expected terminal wealth level depends only on current spot price s (as the inventory level
information has been included in Wt ).
By the law-of-total-variance (e.g., Weiss 2005), the variance of the terminal wealth is given by
[
]
T +1
∑
α−(T +1−i) [πi∗ + αHi (Si+1 )]
V art Wt + α−(T +1−t) [πt + αHt (St+1 )] +
[
i=t+1
[
= Et V art+1 α
−(T +1−t)
[πt + αHt (St+1 )] +
T
∑
α
−(T +1−i)
]]
[π + αHi (Si+1 )]
∗
i
i=t+1
[
[
]]
T
∑
+ V art Et+1 α−(T +1−t) [πt + αHt (St+1 )] +
α−(T +1−i) [πi∗ + αHi (Si+1 )]
[
= Et V art+1
[ ∑
T
i=t+1
]]
α
−(T +1−i)
∗
i
[π + αHi (Si+1 )]
i=t+1
[
[ ∑
]]
T
+V art α−(T +1−t) [πt + αHt (St+1 )] + Et+1
α−(T +1−i) πi∗
[
= Et V art+1
[
[ ∑
T
i=t+1
i=t+1
]]
α
−(T +1−i)
]
∗
i
[π + αHi (Si+1 )]
[
= Et V art+1 [WT +1 ] + V art α
−(T +1−t)
]]
[
−(T +1−t)
+ V art α
[πt + αHt (St+1 )] + vt+1 (St+1 )
]]
[πt + αHt (St+1 )] + vt+1 (St+1 ) .
Notice that J˜t+1 (Wt+1 , St+1 ) = Et+1 [WT +1 ] − λV art+1 [WT +1 ]. Then the equation (4) holds.
Proof of Proposition 2
Observe that the variance of the terminal wealth conditional on period t does not depend on the initial
wealth Wt in period t and the expected terminal wealth is summation of α−(T +1−t) Wt and vt (s). Replacing
J˜t (Wt , s) by Jt (s) + α−(T +1−t) Wt in (4) yields
{
Jt (s) + α−(T +1−t) Wt =
max
α−(T +1−t) Wt + α−(T +1−t) Et [πt ] + Et [Jt+1 (St+1 )]
yt ≥0,Ht ∈Ht
}
[ −(T +1−t)
]
−λV art α
[πt + αHt (St+1 )] + vt+1 (St+1 ) .
Subtracting α−(T +1−t) Wt from both sides of above equation leads to
{
Jt (s) =
max
Et [Jt+1 (St+1 )]
yt ≥0,Ht ∈Ht
]}
[
[
]
,
+α−(T +1−t) Et [πt ] − α−(T +1−t) λV art πt + αHt (St+1 ) + αT +1−t vt+1 (St+1 )
{
= max Et [Jt+1 (St+1 )]
yt ≥0
[
]}
{
[
]}
−(T +1−t)
−(T +1−t)
T +1−t
+α
Et [πt ] − α
λ min V art πt + αHt (St+1 ) + α
. (28)
vt+1 (St+1 )
Ht ∈Ht
Note that dt and St+1 are independent for any given St = s. Applying the law of total variance yields
V art [πt + αHt (St+1 ) + αT +1−t vt+1 (St+1 )]
= Et [V art [πt + αHt (St+1 ) + αT +1−t vt+1 (St+1 )|St+1 ]] + V art [Et [πt + αHt (St+1 ) + αT +1−t vt+1 (St+1 )|St+1 ]]
33
= Et [V art [πt |St+1 ]] + V art [(rt − s)y − (rt + ht − αSt+1 )Et [(y − dt )+ ] + αHt (St+1 ) + αT +1−t vt+1 (St+1 )]
= Et [(rt + ht − αSt+1 )2 V art [(y − dt )+ ]] + V art [αSt+1 Et [(y − dt )+ ] + αHt (St+1 ) + αT +1−t vt+1 (St+1 )]
= Et [(rt + ht − αSt+1 )2 ]V art [(y − dt )+ ] + V art [αSt+1 Et [(y − dt )+ ] + αHt (St+1 ) + αT +1−t vt+1 (St+1 )]. (29)
Here the first equality applies the law of total variance directly, the second is due to the fact that the payoff
of the hedge, αHt (St+1 ), becomes a constant when St+1 is given (conditioned on), and the third equality is
by the fact that
V art [πt |St+1 ] = V art [(rt − s)y − (rt + ht − St+1 )(y − dt )+ |St+1 ] = (rt + ht − St+1 )2 V art [(y − dt )+ ].
Replacing the variance term in the RHS of (28) by the RHS of (29) yields (7).
Proof of Corollary 2
Note that
∂ 2 gt (y, s)
= −2α−(T +1−t) E[(y − dt )+ ]At (s)Φ̄t (y|s) ≤ 0,
∂y∂λ
which implies that gt (y, s) is submodular in (y, λ). By Theorem 2.8.1 of Topkis (1998), yt∗ (s; λ) is decreasing
in λ.
Proof of Corollary 3
First, we know that E[πt (y, s, St+1 )] is concave in y, which implies that it is increasing on the left hand
side of the risk-neutral inventory level yt∗ (s; 0). For any λ > 0, it follows from Corollary 2 that yt∗ (s; λ) is
decreasing in λ, which implies that yt∗ (s; λ) ≤ yt∗ (s; 0). Then, combining the concavity of E[πt (y, s, St+1 )] and
monotonicity of yt∗ (s; λ), we know that E[πt∗ ] is decreasing in λ, which in turn implies that Et [WT∗+1 ] is also
decreasing in λ.
It is obvious that Ut is decreasing in λ for any given policy. Then, Ut∗ must be decreasing in λ. We argue
that V art [WT∗+1 ] must be decreasing in λ. Suppose, otherwise, there exist λ1 > λ2 such that the corresponding
variances, denoted by Vt1 and Vt2 respectively, satisfy Vt1 > Vt2 . Let u1 and u2 be the corresponding optimal
1
2
1
2
policies and WTu+1
and WTu+1
the terminal wealth levels under policies u1 and u2 . Note that WTu+1
≤ WTu+1
.
Then, we have
1
1
2
2
Utu1 = Et [WTu+1
] − λ1 V art [WTu+1
] < Et [WTu+1
] − λ1 V art [WTu+1
] = Utu2 ,
that is policy u2 is strictly better than the optimal policy u1 with respect to λ1 , which leads to a contradiction.
The desired result holds.
A Technical Lemma
The Fundamental Theorem of Calculus states that:
A function f : [a, b] → R is absolutely continuous if and only if it is differentiable almost everywhere, its
derivative f ′ ∈ L1 [a, b] and for any t ∈ [a, b],
∫
t
f ′ (s)ds.
f (t) = f (a) +
a
34
Lemma 2. Suppose that function f : R+ → R is first-order differentiable and f ′ is continuous in (0, ∞),
′′
′
and differentiable in (0, r), |f (r−)| < ∞ and f ≡ 0 in (r, ∞). Then, f ′ is differentiable almost everywhere,
′′
f ∈ L1 [0, ∞) and, for any S > s,
∫
S
f ′ (S) = f ′ (s) +
′′
f (u)du.
s
That is, f ′ is absolutely continuous.
′′
′
Proof. Note that f (s) = 0 for s > r and f is differentiable in (0, r). Then, f ′ is differentiable almost
′′
′′
′′
everywhere. Moreover, |f (r−)| < ∞ and f (r+) = 0, which implies that f ∈ L1 [0, ∞). For any S > s ≥ 0,
if S ≤ r, by the Fundamental Theorem of Calculus, we have
∫ S
′′
′
′
f (S) = f (s) +
f (u)du.
s
If S > r > s,
∫
r
f ′ (S) = f ′ (r) = f ′ (s) +
∫
′′
r
f (u)du = f ′ (s) +
s
∫
′′
S
s
If S > s > r,
∫
′′
r
∫
S
f ′ (S) = f ′ (s) = f ′ (s) +
S
f (u)du = f ′ (s) +
f (u)du +
′′
f (u)du.
s
′′
f (u)du.
s
The desired result holds.
Q.E.D.
Proof of Theorem 2
The proof is by induction. Note that vT +1 (s) = 0. For any t, suppose that vt+1 (·) is twice differentiable almost
everywhere. Then, in a similar way to Carr and Madan (2001), applying Lemma 2 and the Fundamental
Theorem of Calculus, vt+1 (St+1 ) can be decomposed as
′
′
vt+1 (St+1 ) = [vt+1 (s) − vt+1 (s)s] + vt+1 (s)St+1
∫ s
∫
′′
+
vt+1 (K)(K − St+1 )+ dK +
0
∞
′′
vt+1 (K)(St+1 − K)+ dK.
s
That is, the vt+1 (St+1 ) can be perfectly replicated by bonds, forwards and options. For any y, with complete
term and strike structure, the optimal hedging decision is to offset St+1 Et [(y − dt )+ ] and αT −t vt+1 (St+1 ) so
that ψt (y, s) = 0. That is,
[
′
Ht (St+1 ) = − (Et [(y − dt )+ ] + αT −t vt+1 (s))[St+1 − ft (s)]
∫ s
∫
′′
+
αT −t vt+1 (K)[(K − St+1 )+ − Pt (K)]dK +
0
∞
α
T −t ′′
t+1
v
]
(K)[(St+1 − K) − Ct (K)]dK .
+
s
′
That is, given any inventory level y, the optimal hedging strategy is to short Et [(y − dt )+ ] + αT −t vt+1 (s)
′′
′′
units of forwards, αT −t vt+1 (K)dK puts for all strikes K < s and αT −t vt+1 (K)dK calls for all strikes K > s
maturing at next period. Then,
gt (y, s) = Et [πt ] − λα−(T +1−t) At (s)V ar[(y − dt )+ ].
Taking derivative with respect to y yields
∂gt (y, s)
= (rt − s) − (rt + ht − αft (s))Φt (y|s) − 2λα−(T +1−t) E[(y − dt )+ ]At (s)Φ̄t (y|s),
∂y
[
]
= rt + ht − αft (s) − 2α−(T +1−t) λE[(y − dt )+ ]At (s) Φ̄t (y|s) − (s + ht − αft (s)).
(30)
35
Note that both factors of of the first term in the RHS of the second equality are decreasing in y. By
Assumption 1, s + ht − αft (s) ≥ 0. If the first term is negative at some point y 0 , then gt (y, s) is decreasing
in y ≥ y 0 . This implies that the optimal point yt∗ ≤ y 0 . When the first term is nonnegative, it is clear that
∂gt (y,s)
∂y
is decreasing in y, i.e., gt (y, s) is concave in y. More precisely, it is strictly concave in y. Then, gt (y, s)
is unimodal in y for any s. Note that limy→0+
∂gt (y,s)
∂y
= rt − s. Then, if rt > s, yt∗ (s) is the unique maximizer
of gt (·, s) which satisfies (17). If rt ≤ s, the unimodularity of gt implies that gt is decreasing in y, which
implies that the least maximizer yt∗ (s) = 0.
Notice that all the terms of the RHS of (30) are differentiable in s, which implies that
∂gt (y,s)
∂y
is also
differentiable in s.
Taking derivatives on both sides of (30) with respect to s and y, respectively, and letting y = yt∗ (s), we
have
∂ 2 gt (yt∗ (s), s)
′
= −1 + αft (s)Φt (yt∗ (s), s) − (rt + ht − αft (s))ϕt (yt∗ (s), s)
∂y∂s
−2λα−(T +1−t) E[(yt∗ (s) − dt )+ ][A′t (s)Φ̄t (y|s) − At (s)ϕt (yt∗ (s), s)],
2
∗
[
]
∂ gt (yt (s), s)
= − (rt + ht − αft (s)) − 2λα−(T +1−t) E[(yt∗ (s) − dt )+ ]At (s) ϕt (yt∗ (s), s)
∂y 2
−2λα−(T +1−t) At (s)Φt (yt∗ (s))Φ̄t (yt∗ (s), s)
s + ht − αft (s)
ϕt (yt∗ (s), s) − 2λα−(T +1−t) At (s)Φt (yt∗ (s))Φ̄t (yt∗ (s), s),
=−
Φ̄t (yt∗ (s), s)
(31)
(32)
where the second equality is by the first-order condition.
When s < rt , we know that
∂ 2 gt (yt∗ (s),s)
∂y 2
< 0. This implies that yt∗ (s) is differentiable in s and satisfies
∂ 2 gt (y ∗ (s),s)
t
dyt∗ (s)
∂y∂s
= − ∂ 2 gt (y∗ (s),s) .
t
ds
2
∂y
It is clear that
∂ Φt (yt∗ (s),s)
∂y
is differentiable in s since Φt is twice differentiable and yt∗ (s) is differentiable.
Then, all the terms of the RHS of (31) are differentiable. Similarly, all the terms of the RHS of (32) are
differentiable. Then
dyt∗ (s)
ds
is differentiable in s < rt , i.e., yt∗ (s) is twice differentiable in s < rt .
When s > rt , we have shown that yt∗ (s) = 0, which implies that the first- and second-order derivatives of
yt∗ (s) are both zero as s > rt . However, as rt = s = αft (s) − ht ,
∂ 2 gt (yt∗ (s),s)
∂y 2
= 0 which implies that yt∗ (s) may
not be differentiable at the point s = rt . Thus, yt∗ (s) is twice differentiable in s almost everywhere.
Note that
∫
Et [πt ] = (rt − s)y − (rt + ht − αft (s))
y
Φt (ξ, s)dξ
0
is twice differentiable in s by Assumption 3 since ft (s) = E[St+1 |St = s] and Φt (ξ, s) are twice differentiable
in s. Let ψt (s) = Et [πt (yt∗ (s), s, St+1 )]. We next show that ψt (s) is twice differentiable in s almost everywhere.
First, for s < rt
∂ψt (s)
= −yt∗ (s) + αft′ (s)
∂s
∫
yt∗ (s)
∫
yt∗ (s)
Φt (ξ, s)dξ − (rt + ht − αft (s))
0
+[(rt − s) − (rt + ht − αft (s))Φt (yt∗ (s), s)]
ϕt (ξ, s)dξ
0
∂yt∗ (s)
.
∂s
Since yt∗ (s) is twice differentiable in s < rt , then Et [πt (yt∗ (s), s, St+1 )] is twice differentiable in s < rt .
36
When s > rt , we know that yt∗ (s) = 0, which infers that ψt (s) = Et [πt (yt∗ (s), s, St+1 )] = 0, and therefore,
ψt (s) is twice differentiable in s for s > rt . It is obvious that lims→rt +
′
t
∂ψt (s)
∂s
= 0. Then, ψt (s) is differentiable
′
t
in s for all s > 0 and ψ (s) is continuous in s with ψ (rt ) = 0.
Thus, ψt (s) = Et [πt (yt∗ (s), s, St+1 )] is twice differentiable in s almost everywhere.
By induction assumption that vt+1 (s) is twice differentiable in s almost everywhere and Assumption (3),
we know that Et [vt+1 (St+1 )] is twice differentiable in s almost everywhere as well. By recursive equation (5),
we know that vt (s) is twice differentiable in s almost everywhere as well. This completes the induction.
Proof of Theorem 1
For any t, let Ht (St+1 ) = −Et [(y − dt )+ ](St+1 − ft (s)) + H̃t (St+1 ). Then, we have
ψ(y, s) = max {V art [αT −t vt+1 (St+1 ) + H̃t (St+1 )]},
H̃t ∈Ht
which is independent of y. Then, we have
]
∂gt (y, s) [
= rt + ht − αft (s) − 2α−(T +1−t) λE[(y − dt )+ ]At (s) Φ̄t (y|s) − (s + ht − αft (s)).
∂y
In a similar way of the Proof of Theorem 2, we can show Parts (a) and (b).
If only forward contracts maturing at next period are available, letting H̃t = q[St+1 − ft (s)] and solving
min V art [q[St+1 − ft (s)] + αT −t vt+1 (St+1 )]
q
yields the optimal solution
qt∗ (s) = −
Covt [St+1 , αT −t vt+1 (St+1 )]
.
V art [St+1 ]
Proof of Proposition 3
Note that vT +1 (s) = 0. Taking first order derivative for ĝT (y, s) yields
[
]
∂ĝT (y, s)
= rt + hT − αfT (s) − 2α−1 λE[(y − dT )+ ]AT (s) Φ̄T (y, s) − (s + hT − αfT (s))
∂y
−2λαV arT [ST +1 ]ΦT (y, s)ET [y − dT ]+ .
Observe that the first and the third terms of the RHS of above equation are decreasing in y. Similar to the
proof of Theorem 2, we can show that ĝT (y, s) is unimodual in y.
When rt ≤ s, ĝT (y, s) is decreasing in y and thus ȳT∗ (s) = 0. Otherwise, ĝT (y, s) is unimodual in y and the
optimal solution satisfies (21). By Theorem 2 and Theorem 1, the hedged optimal inventory decision yT∗ (s)
satisfies
[
]
rt + hT − αfT (s) − 2α−1 λE[(y − dT )+ ]AT (s) F̄T (y, s) − (s + hT − αfT (s)) = 0.
This implies that
∂ĝT (yT∗ , s)
= −2λαV arT [St+1 ]FT (yT∗ (s), s)ET [yT∗ (s) − dT ]+ < 0.
∂y
The unimodularity of ĝT (y, s) in y implies that ŷT∗ (s) < yT∗ (s) as rt > s.
37
Proof of Proposition 4
Differentiating ĝt with respect to y yields
[
]
∂
gt (y, s) = rt + ht − αft (s) − 2α−(T +1−t) λE[(y − dt )+ ]At (s) Φ̄t (y|s) − [s + ht − αft (s)]
∂y
[
]
−(T −t−1)
+
T −t Covt (St+1 , vt+1 (St+1 ))
−2α
Φt (y|s)V art (St+1 ) E[(y − dt ) ] + α
.
V art (St+1 )
(33)
Not that the sum of the first two terms of RHS of (33) is strictly positive if 0 < y < yt∗ while the third term
is decreasing in y. If E[(yt∗ (s) − dt )+ ] < −αT −t
implies that
∂
∂y
implies that
, then the third term is positive, which
ĝt (y, s) > 0 as y ≤ y and therefore the optimal solution ŷt∗ (s) > yt∗ (s). If, on the contrary,
∗
t
E[(yt∗ (s) − dt )+ ] ≥ −αT −t
∂
∂y
Covt (St+1 ,vt+1 (St+1 ))
V art (St+1 )
Covt (St+1 ,vt+1 (St+1 ))
V art (St+1 )
, then the third term is negative and decreasing in y, which
ĝt (y, s) ≤ 0 as y ≥ yt∗ and therefore the optimal solution ŷt∗ (s) ≤ yt∗ (s).
Proof of Proposition 5
Let z = y − dt . Then,
gt (dt + z, s) = (rt − s)(dt + z) − (rt + ht − αft )z + − λα−(T −1−t) V art [St+1 z + + αT −t vt+1 (St+1 )].
When z > 0, we have
∂
gt (z, s) = −(s + ht − αft ) − 2λα−(T −1−t) V art (St+1 )[z + αT −t Covt (St+1 , vt+1 (St+1 ))/V art (St+1 )],
∂z
which is decreasing in z. This implies that gt (dt + z, s) is concave in z as z > 0.
It is clear that for any s,
∂
∂z
gt (z, s) → −∞ as z → ∞. When z = 0,
∂
gt (dt , s) = −(s + ht − αft ) − 2λαCovt (St+1 , vt+1 (St+1 )).
∂z
−αft
If Covt (St+1 , vt+1 (St+1 )) < − s+h2tλα
, then
∂
∂z
gt (dt , s) and the optimal inventory decision must be greater
than dt ; otherwise, the optimal inventory decision must be less than dt .
Proof of Theorem 3
First, the inequalities of part (a) follow directly from Lemma 1 since the inequalities can be preserved under
the minimization operations. Part (a) holds.
Note that the optimal lagrange multipliers must be nonnegative. We argue that they must be positive as
δ < δ0 . If, otherwise, λ∗i = 0, the resulted variance must be equal to δ0 , which violates the constraint. By
KKT theorem, a positive lagrange multiplier implies that the constraint is binding. Then, by part (a), we
know that the inequalities of part (b) hold.
Note that the objective function of the lagrange problem is supermodular in (λ, δ). Then, under the
minimization operation with respect to λ, the supermodularity implies that λ∗i must be decreasing in δ. Since
E[WTi +1 ] is decreasing in λ, we know that E[WT∗i+1 ] is increasing in δ. It is clear that as δ → δ0 , E[WT∗i+1 ]
converges to risk-neutral expected terminal wealth. Part (c) holds.
Recall that for any λ, E0 WTe +1 = E0 WTf +1 but V ar0 [WTe +1 ] ≤ V ar0 [WTf +1 ] (by Lemma 1). Then, when
λ = λ∗f , E0 WTe +1 = E0 WT∗f+1 but V ar0 [WTe +1 ] ≤ V ar0 [WT∗f+1 ] = δ. By Corollary 3, we know that to bind the
variance constraint, one must lower the value λ, i.e., λ∗e ≤ λ∗f . Applying Corollary 2, we know that yt∗e ≤ yt∗f .
Part (d) holds.